CPP
1. Find the derivative of
2
1
(i) x (ii) x2ex (iii) x (x + 1) (x + 2)
x
x2 3 1 x x 1
(iv) (v) ,x1 (vi) cos log
x 4 1 x x
x2
(vi) sin1 (3x 4x3) (vii) log sin 1 .
3
dy
(2) Find , when
dx
...
x...
xx
(i) y= x x x ....... (ii) y xx
dy
3. Find , when
dx
(i) x = a ( + sin ), y = a (1 + sin ) (ii) x = et(sin t + cot t), y = et log t.
t
(iii)x = a cos t log tan , y = a sin t. (iv) x = log (1 + t2), y = tan1 t.
2
4. Find the derivative of
2x 1
(i) 2 , with respect to sin x2, (ii) sin x2 with respect to x2.
x 1
1 x2 1 3x x
3
(iii) cos1 , with respect to tan .
1 x 2 1 3x 2
dy
5. Find , when
dx
(i)x cos y + x2 y3 = 0, (ii) x2 y2 = c (x2 + y2)2 (iii) xy = ex+y.
(iv) sin y = x sin (x + y) (v) (tan1 x)y + ycotx = 1.
d2y
(6) If y = A cos nx + B sin nx, prove that n2 y 0 .
dx 2
d2 y dy
(7) If y = bex + ce2x, prove that 2
3 2y 0 .
dx dx
d2 y dy
(8) If y = a sin(logx) + b cos(logx), prove that x 2 x y 0.
dx 2 dx
Answer Keys
1 x 2 8x 3
1. (a) 1 (b) (x2 + 2x)ex. (c) 3x2 + 6x + 2. (d)
x2 (x 4)2
x2
x cot 1
1 1 x 1 3
(e) (f) sin log (g)
(1 x) 1 x 2 x(1 x) x x2
3 log sin 1
3
1 y2
2. (i) (ii)
2y 1 x(1 y log x)
CPP
cos t log t 1 1
3. (a) (b) (c) tan t (d)
1 cos
t sin t cos t cot t coesc t 2
2t
1 x x2 2
4 (a) 2 2 2
(b) cos x 2 (c)
x(x 1) cos x 3
2x cos y x 3y 2 x 2 xy
5. (a) (b) (c)
2
3y x sin y y 3x y
2 2 x(log x 1)
cot x 2 (tan1 x)y y
y cos ec log y
(d)
sin(x y) x cos(x y) (e) (1 x 2 ) tan1 x
cos y x cos(x y) y cot x 1 cot x (tan1 x)y log tan1(x)
